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Math Help - symmetry

  1. #1
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    symmetry

    Let G be the dihedral group of symmetries of the square (=D_4)

    (a) what is the stabilizer of a vertex? an edge?
    (b) G acts on the set of two elements consisting of the diagonal lines. What is the stabilizer of a diagonal?

    could anyone please help me to solve this problem??
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    Let G be the dihedral group of symmetries of the square (=D_4)

    (a) what is the stabilizer of a vertex? an edge?
    Let 1, 2, 3, and 4 are vertices of the square (counterclockwise order) and e_1, e_2, e_3, and e_4 are edges between these vertices.
    Let G_x be the stabilizer of x. Then,

    G_1 = {e, (2,4) }, G_2 = {e, (1,3)}, ....
    G_e1 = {e, (1,2)(3,4)}, G_e2 = {e, (14)(23)}, ....


    (b) G acts on the set of two elements consisting of the diagonal lines. What is the stabilizer of a diagonal?
    The stabilizer elements for the above action are identity, 180 degree rotation, diagonal flip, and 180 degree rotation after diagonal flip. What are these elements in D_4?
    Last edited by aliceinwonderland; October 24th 2009 at 01:24 PM. Reason: correction
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